Problem: $\sum\limits_{k=1}^{900 }{{(4k + 6)}}=$
What is the question asking for? The question is asking for the sum of the values of $4k + 6$ from $k = 1$ to $k = 900$ : $(4 \cdot 1 + 6) + (4 \cdot 2 + 6) +... + (4\cdot {900} +6)$ The series is arithmetic because the formula $4k + 6$ is a linear function of $k$. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The number of terms $(n = {900})$ is the upper limit of the sigma notation. We need to find $a_1$ (the first term) and $a_{900}$ (the last term). Step 1: Find $a_1$ and $a_{900}$ (the first and the last term) $a_1 = 4(1) + 6 = {10}$ $a_{900} = 4(900) + 6 = {3606}$ Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{900}} &= \dfrac {({10} + {3606})}2 \cdot {900} \\\\ S_{900} &= 1808(900) \\\\ S_{900} & = 1{,}627{,}200 \end{aligned}$ The answer $1{,}627{,}200$